6.7 Notes – Inverse Functions

Bellwork:On graph paper, sketch the following functions on the SAME set of axes:1) 2) 3) 4)

What do you notice about the graphs?

Homework:Read 6.76.7 Part 1 (1-3, 12-20, 42-47)6.7 Complete Solutions - Part 1 (in documents)

𝑦=𝑥

𝑦=𝑥2+2

𝑦=𝑥

𝑦=𝑥2+2

𝑦=𝑥

𝑦=𝑥2+2

𝑦=√𝑥−2

𝑦=𝑥

𝑦=𝑥2+2

𝑦=√𝑥−2

𝑦=√𝑥−2

𝑦=𝑥

𝑦=𝑥2+2

𝑦=−√𝑥−2

𝑦=√𝑥−2

𝑦=𝑥

𝑦=𝑥2+2

𝑦=−√𝑥−2

𝑦=√𝑥−2

𝑦=𝑥𝑦=𝑥2+2

𝑦=−√𝑥−2If you reflect the graph of across the line , you end up with the graph for !

Reflecting across line is the same as just swapping the x-values and y-values!

If you reflect the graph of across the line , you end up with the graph for !

Reflecting across line is the same as just swapping the x-values and y-values! 𝑦=𝑥2+2 𝑦=√𝑥−2 𝑦=−√𝑥−2

Notice how the x-y values are reversed for the original function and the reflected functions.

Definitions:Inverse Relations – Relations that UNDO eachother.

The graphs of inverse relations are reflections of each other across the line

The coordinates for x and y are swapped when going between a relation and its inverse.

Review from chapter 2

• Relation – a mapping of input values (x-values) onto output values (y-values).

• Here are 3 ways to show the same relation.

y = x2 x y

-2 4

-1 1

0 0

1 1

Equation

Table of values

Graph

• Inverse relation – just think: switch the x & y-values.

x = y2

xy

x y

4 -2

5 -1

0 0

1 1

** the inverse of an

equation: switch the x

& y and solve for y. ** the

inverse of a table:

switch the x & y.

** the inverse of a graph: the reflection of the original graph

in the line y = x.

Ex: Find an inverse of y = -3x+6.• Steps: -switch x & y

-solve for y

y = -3x+6

x = -3y+6

x-6 = -3y

yx

3

6

23

1 xy 2

3

11 xy

Inverse Functions

• Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other.

Symbols: f -1(x) means “f inverse of x”

Ex: Verify that f(x)=-3x+6 and g(x)=x+2 are inverses.

• Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.

f(g(x))= -3(x+2)+6

= x-6+6

= x

g(f(x))= (-3x+6)+2

= x-2+2

= x

** Because f(g(x))=x and g(f(x))=x, they are inverses.

To find the inverse of a function:

1. Change the f(x) to a y.

2. Switch the x & y values.

3. Solve the new equation for y.

4. Call y what it actually is: .

** Remember functions have to pass the vertical line test!

Ex: (a)Find the inverse of f(x)=x5.

1. y = x5

2. x = y5

3. 5 55 yx

yx 5

5 xy

(b) Is f -1(x) a function?

(hint: look at the graph!

Does it pass the vertical line test?)

Yes , f -1(x) is a function.

5 x

Horizontal Line Test

• Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test.

• If the original function passes the horizontal line test, then its inverse is a function.

• If the original function does not pass the horizontal line test, then its inverse is not a function.

Ex: Graph the function f(x)=x2 and determine whether its inverse is a

function.

Graph does not pass the horizontal line test, therefore the inverse is not a function.

Ex: f(x)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.

2

2

4y

x

f -1(x)= is not a function.

y = 2x2-4

x = 2y2-4

x+4 = 2y2

2

4x

y

22

1 xyOR, if you fix the

tent in the basement…

22

1 x

Ex: g(x)=2x3

Inverse is a function!

y=2x3

x=2y3

3

2y

x

yx

3

2

3

2

xy

OR, if you fix the tent in the basement…

2

43 xy

31

2)(

xxg